To find [tex]\(\tan 45^\circ\)[/tex], we need to use some fundamental concepts from trigonometry. Angle measures in trigonometry can be represented in both degrees and radians. For this problem, consider the angle in degrees.
1. Understanding the Tangent Function:
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. For any angle [tex]\(\theta\)[/tex],
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
2. Special Angles:
Certain angles have well-known tangent values. One such angle is [tex]\(45^\circ\)[/tex]. For an angle of [tex]\(45^\circ\)[/tex], the opposite and adjacent sides of a right triangle are equal.
3. Unit Circle and Trigonometric Ratios:
On the unit circle, [tex]\(\tan \theta\)[/tex] is the y-coordinate divided by the x-coordinate for any angle [tex]\(\theta\)[/tex], ensuring the point on the circle has coordinates ([tex]\(\cos(\theta), \sin(\theta)\)[/tex]). For [tex]\(\theta = 45^\circ\)[/tex], both sine and cosine values are equal:
[tex]\[
\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}
\][/tex]
Thus,
[tex]\[
\tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1
\][/tex]
4. Result:
Therefore, the tangent of [tex]\(45^\circ\)[/tex] is:
[tex]\[
\tan 45^\circ = 1
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
